Franco Ramirez

Nonlinear analysis tools for the optimized design of harmonic-injection dividers

New nonlinear analysis tools for harmonic-injection dividers are presented based on bifurcation concepts. The advantage of these tools is their application simplicity and efficiency, which has enabled their use for actual circuit design and optimization. The tools allow control over the divided frequency and output power and predict the variation of the synchronization bands versus the circuit element values, which facilitates design correction. They have been extended to the analysis and optimization of phase-locked harmonic-injection dividers, which contain a low-frequency feedback loop. The use of this loop, together with the accuracy of the analysis, has enabled the implementation of novel frequency-division functions, such as the division of variable order, versus a circuit parameter, or the division by fractional order. The output noise of the frequency dividers is analyzed through the conversion-matrix approach, studying the noise variation along the division bands. The new techniques have been applied to the design of a frequency divider by order 4 and 5, with 18-GHz input frequency, and excellent agreement with experimental results has been obtained.

Phase-Noise Analysis of Injection-Locked Oscillators and Analog Frequency Dividers

In-depth investigation of the phase-noise behavior of injection-locked oscillators and analog frequency dividers is presented. An analytical formulation has been obtained, which allows a better understanding of the shape of the output phase-noise spectrum of these circuits. The simplicity of this formulation is also helpful for circuit design. Approximate expressions for the corner frequencies of the spectrum are determined, identifying the most influential magnitudes and deriving design criteria. In particular, a technique has been developed to shift the frequency of the first corner of the phase-noise spectrum, up to which the output phase noise follows the input one. The expressions for the corner frequencies can be introduced in either in-house or commercial harmonic-balance software, thus allowing an agile design, as no separate phase-noise analysis is required. The validity of the analytical techniques is verified with the conversion-matrix approach and with measurements using two field-effect-transistor-based circuits: a 4.9-GHz injection-locked oscillator and a frequency divider by 2 with 9.8-GHz input frequency.

Phase and Amplitude Noise Analysis in Microwave Oscillators Using Nodal Harmonic Balance

In this paper, a nodal harmonic balance (HB) formulation is presented for the phase and amplitude noise analysis of free-running oscillators. The implications of using different constraints in the resolution of the perturbed-oscillator equations are studied. The obtained formulation allows the prediction of the possible spectrum resonances without ill conditioning at low frequency offset from the carrier. The noise spectrum is meaningfully expressed in terms of the eigenvalues of a newly defined matrix, obtained from the linearization of the nodal HB system about the steady-state solution. The cases of real or complex-conjugate dominant eigenvalues are distinguished. The developed phase-noise formulation is extended to a system of two coupled oscillators. The phase and amplitude noise analyses have been applied to a push-push oscillator at 18 GHz, a bipolar oscillator at 1 GHz, and a coupled system of two field-effect transistor oscillators at 6 GHz.

Harmonic-balance analysis and synthesis of coupled-oscillator arrays

A new harmonic-balance technique for the analysis and synthesis of coupled-oscillator systems with adjustable phase progression is presented here. It allows an accurate determination of the phase-shift versus the tuning elements, since detailed models are used for the individual oscillator circuits. It is also possible to impose the synchronization frequency at which the in-phase oscillation takes place and calculate the values of the tuning elements for a pre-fixed phase distribution. The phase noise of the system is analyzed through the carrier modulation approach. The phase-change rapidity versus a control voltage is analyzed through the envelope-transient method. A four-element oscillator array at 5 GHz has been designed through this method and experimentally characterized obtaining very good agreement with the simulation results.

Analysis of Near-Carrier Phase-Noise Spectrum in Free-Running Oscillators in the Presence of White and Colored Noise Sources

This paper presents the stochastic characterization of the phase-noise spectrum of free-running oscillators in the envelope domain. The flicker-noise sources are modeled as an infinite summation of Ornstein-Uhlenbeck processes. Inputs of the formulation are parameters extracted with harmonic balance (HB) by means of the carrier-modulation approach. These parameters can be easily identified with commercial software packages without user access to the Jacobian matrix of the HB system. The presented envelope-domain formulation significantly reduces the complexity of the stochastic analysis in comparison with time-domain techniques. The phase-noise spectrum depends on the measurement time interval, instead of estimated cutoff frequencies for the colored noise sources. With the new formulation it has been possible to investigate the influence of the measurement time interval on the near-carrier phase-noise spectrum. The formulation will show that provided the measurement time fulfills some nonstrict mathematical conditions, typically accomplished in any practical measurement, the full phase-noise spectrum will be nearly the same, whatever the measurement time interval. Analytical expressions are provided for the full phase-noise spectrum. All the obtained expressions have been exhaustively validated by means of Monte Carlo simulations. In order to verify the generality of the results, the analysis method has been applied to various oscillator configurations. The simulated and measured phase-noise spectra have been compared, obtaining very good agreement in all cases.

Stability and Noise Analysis of Coupled-Oscillator Systems

We present a simplified closed-form formulation for the optimized design of coupled-oscillator systems. It is based on admittance models for the oscillator elements, extracted from HB simulations. The formulation relates explicitly the coupled-system oscillation frequency and amplitude and phase distributions with the parameters of the coupling network and the oscillator elements. It allows anticipating and understanding the form of variation of the system variables and can be used for an insightful design. With a perturbation analysis based on the new formulation, we will generalize existing stability criteria to more complete oscillator models. A combined amplitude- and phase-noise formulation will enable the prediction of the phase-noise spectrum flattening near oscillator carrier, while taking into account the system resonances that affect the spectrum shape. The techniques have been successfully applied to a coupled-oscillator system at 5.2 GHz.

Stability and Bifurcation Analysis of Self-Oscillating Quasi-Periodic Regimes

An in-depth stability and bifurcation analysis of self-oscillating quasi-periodic solutions is presented. It is based on the formal analysis of the frequency-domain characteristic system, with a high degree of complexity due to the repetition of singularities at the intermodulation frequencies of the quasi-periodic spectrum. The problem is tackled by relating the system singularities to the Lyapunov exponents so that equivalent singularities of the frequency-domain system are mapped into the same Lyapunov exponents. The study is illustrated by means of its application to a self-oscillating power amplifier, which is used here as a test bench. The main types of qualitative behavior versus relevant circuit parameters, such as the bias voltage and input power, are distinguished and analyzed in detail. The influence of the transistor biasing on the number of oscillatory solutions is studied, as well as the effect of these coexisting solutions on the circuit response versus the input power. Two types of hysteresis are identified and explained, as well as a co-dimensional 2 bifurcation, which leads to a qualitative change in the structure of the quasi-periodic solution curves. The analysis is validated with measurement results.

Analysis of stabilization circuits for phase-noise reduction in microwave oscillators

Two configurations for oscillator phase-noise reduction using stabilization circuits have been demonstrated in the literature. One of them is based on the self-injection of the oscillator signal, after it passes through a long delay line or a high-quality-factor resonator. The second one is a stabilization loop, containing a frequency discriminator. In this paper, an in-depth analytical comparison of these two configurations, respectively based on injection locking and phase-locking principles, is presented. Analytical expressions are provided for the variation of the steady-state solution and its phase noise versus the parameters of the feedback network. The expressions are rigorously validated with harmonic balance. Instabilities reported by other authors are investigated through bifurcation analysis. The new expressions enable a good understanding of the amplitude and frequency jumps and sharp phase-noise maxima obtained simulations and measurements versus the feedback parameters. A practical 5-GHz voltage-controlled oscillator has also been implemented, for validation purposes.

General stabilization techniques for microwave oscillators

A general stabilization technique is proposed to suppress undesired spurious oscillations in microwave oscillators. The purpose is to eliminate these oscillations while maintaining the oscillation frequency and amplitude of the originally-unstable solution. The main advantage of the technique is its wide generality of application, not restricted to low-frequency spurious oscillations. It has been tested on an unstable 18-GHz push-push oscillator that has been manufactured and measured, with very good agreement with the simulation results.

Stability Analysis of Oscillation Modes in Quadruple-Push and Rucker's Oscillators

Oscillator systems composed by N sub-oscillators coupled through a symmetric linear network enable the combination of output power at the first or Nth harmonic component of the oscillation frequency of each sub-oscillator. However, they have the drawback of a possible coexistence of different oscillation modes, which limits their practical application. This paper presents an in-depth stability analysis of coexisting steady-state solutions in Ruckers oscillator and N -push oscillators. Criteria are provided to avoid undesired oscillation modes. The coupled system is described by means of a semianalytical formulation based on numerical models of the active subcircuits, extracted from harmonic-balance (HB) simulations. Each active subcircuit is composed by the transistor(s), feedback elements, and termination load. The use of the HB numerical models allows a realistic prediction of the behavior of the globally coupled oscillator. Alternatively, a graphical technique is provided to obtain the different oscillation modes using full HB simulations. The perturbation of the reduced-order nonlinear system enables the stability and phase-noise analysis of the steady-state oscillatory solutions. In the derived formulation for phase-noise analysis, both flicker- and white-noise perturbations are considered. The different techniques have been applied to a Ruckers oscillator at 3.9 GHz and a quadruple-push oscillator at 15.6 GHz.